Welcome to a video on how to determine the domain of a vector valued function
The domain of a vector valued function is the intersection of the domain of: f of t, g of t and h of t.
When the vectors expressed, as we see here, as a sum of the unit vectors i,j,k.
Or as we see here expressed in component form.
Of course, if the vector valued function only has two components: the x component and the y component.
We'll only be looking for the intersection f of t and g of t. Let's go and take a look at a couple of examples.
Here we want to find the domain of the vector-valued function r of t.
There's only two components we have to find the domain of natural log t plus one.
For the domain of the squared off or minus T squared.
And then determine the intersection of those two domains.
Now if you know a lot about your basic functions it helps find the domain.
But I provided the graphs have these functions to help us determine the domain.
Remember the domain refers to all the possible values of t in the given functions.
Where t would be the values along we normally call the x axis.
So were really asking ourselves how does this function behave moving from left to right.
So we can see here that the domain of natural log T plus one. Looks like it goes from -1 to positive infinity.
Which is true.
But it does not include -1, because if t is -1, we have natural log 0, which is undefined.
So the domain for natural log t plus one, would be t greater than -1.
Or if we want to expresses this using interval notation it would be, open on negative one to positive infinite.
And for the square root of four minus T-square. We know the number underneath the square root, must be greater than or equal to zero.
So we could set this up and solve it as an equation, where four minus t squared must be greater than or equal to zero.
But if t is equal to 2 or -2, we would have 0 greater than or equal to 0 which is true.
So looking at the graph these two end points would be included in the domain.
Of this red graph, so the domain for this function would be where t is;
greater than or equal to -2 and less than or equal to 2
or interval notation would be on the closed intervals from -2 to 2.
Now we want a graph these two intervals on the same number line and determine where they intersect.
So for t greater than -1 we have an open point on -1 arrow to the right.
And for this interval we have to close points one on -2 and one on positive 2.
And the interval is between, so it would look something like this:
The intersection would be the values where's graphed both in blue and in red.
So we can see from the graph the domain have the given vector valued function would be:
The interval from -1 to positive 2 where it is open on -1 and close on positive 2
Or we could say where t is greater than -1.
And less than or equal to positive 2. Let's take a look at another example.
Here we have an x,y and z components. So now we need to determined the domain of all three of these functions.
And then determine where they intersect.
So the first function is natural log t. Again we can see it graphed here:
Again t can't equal 0 but t can be any value larger than 0. So we have t greater than 0.
Using interval notation would have the interval from zero to infinity. Where it's open on zero.
Now for the square root of t plus 2. again we know that t plus 2 must be,
greater than or equal to 0. So we could subtract two on both sides to get t,
greater than or equal to -2. We can also see that here from the graph.
So we have t greater than or equal to -2.
Or we could express it as closed on -2 to positive infinity.
Then our last function we have cosine t.
And if you look at the graft the graph it is moving left and right for ever with no breaks her holes.
That tells us that the domain Cosine t would be all real numbers.
Or from negative infinity to positive infinity.
Let's go ahead and graph of three of these and see where they intersect.
So in blue we have t greater than 0.
Look like this: Next we have t greater or equal to negative 2.
Would look like this: And then in green we have all real numbers which would be the entire number line.
So were looking for all the numbers that are graphed in blue red and green.
And you can see that it happens when t is greater than 0.
Therefore the domain of the vector valued function.
Is t greater than 0 or using interval notation we can express it from 0 to positive infinity, where it is open on 0.
Okay that's going to do it for this video. I hope you found it helpful!